P1. 24 sí y solo sí 48

$\angle ABD = \angle ADB \Rightarrow \angle BAC = 180^{\circ}-2 \angle ADB = 180^{\circ}-\angle ABC - \angle ACB \Rightarrow 2\angle ADB = \angle ABC + \angle ACB$. Así mismo, $\angle ADB=\angle DBC + \angle ACB$. 

 

Entonces $\angle DBC = 24^{\circ} \iff \angle ADB = 24^{\circ} + \angle ACB = \angle ABD \iff \angle ABC = \angle 48^{\circ} + \angle ACB \iff \angle ABC - \angle ACB = 48^{\circ} + \angle ACB - \angle ACB = 48^{\circ}$.

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